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Algebra
http://www.purplemath.com/modules/index.htm
Factoring
self-test 1: 4 * x4 – 4 = ?
Quadratic
The quadratic equation
a*x2 + b* x +c = 0 has the solution
if a is not 0
if c is not 0
Exponents
self-test 2: (312)3 / 924 = 3?
Example: 1060*1040/1058 = 1042
Logarithm
y = logb(x)
is equivalent to
x = by
self-test 3: log16(4096) = ?
log8(4096) = ?
The base b must be neither 0 nor 1, and is typically 10, e, or 2.
Example: since

Example: what is log4(625)? = 5
A common use of log is ln(expx) = x = loge(ex)
Similarly, log10(10x)=x
red base e, green base 10, purple base 1.7
self-test 4: log3(27/81) = ?
For any other base b, we use
Example:
log10 (1,000/10,000) = log(1000) – log(10,000) = 3 – 4 = - 1 = log (1/10)
Factorial
n! = 1 * 2 * 3 * … * n
eg
3! = 1 * 2 * 3 = 6
4! / 6! = 1 / (5*6) = 1 / 30
self-test 5: (n-1)! / (n+1)! = ?
Geometric Concepts
self-test 6 :
1.
The sum of the measures of the interior angles of a triangle is 180°.
In the figure above,
.
2. When two lines intersect, vertical angles are congruent.
In the figure above,
.
3. A straight angle measures 180°.
In the figure above,
.
Area and Perimeter
Volume
Volume of a sphere
(r is the radius of the sphere)
example: cube l = 1cm, cylinder r= 1 cm, h = 4cm,
sphere r = 1 cm – which volume is largest?
= (4/3) r3
self-test 7 :How large do you have to make the
cube length to get the same volume as for the
sphere?
Coordinate Geometry
self-test 8: show the points
(2,4) , (-1,1), and (1,-1)
Slope
self-test 9: draw a graph of
a straight line with slope 1/3
and one with slope 2.5
A line that slopes upward as you go from left to right has a positive slope. A line
that slopes downward as you go from left to right has a negative slope. A
horizontal line has a slope of zero. The slope of a vertical line is undefined.
The equation of a line can be expressed as
is the y-intercept.
The equation of a parabola can be expressed as
vertex of the parabola is at the point
and
If
upward; and if
the parabola opens downward.
where m is the slope and b
where the
the parabola opens
self-test 10: sketch the parabolas
y1 = (x+2)2
y2 = - (x-2)2 + 4
The parabola above has its vertex at
Therefore,
and
The
equation can be represented by
Since the parabola opens downward, we know that
To find the value of a,
you also need to know another point on the parabola. Since we know the parabola
passes through the point
so
Therefore, the
equation for the parabola is
The number of degrees of arc in a circle is 360.
The sum of the measures in degrees of the angles of a triangle is 180.
Trigonometry
http://demonstrations.wolfram.com/IllustratingTrigonometricCurvesWithTheUnit
Circle/
Arcsine
arcsin (sin () =  etc. for cos, tan, cot
asin
sin-1
self-test 11: atan ( tan ( sin (asin () ) ) ) = ?
Definition Domain of x for real result Range
arcsine y = arcsin(x) x = sin(y) −1 to +1 −π/2 ≤ y ≤ π/2
arccosine y = arccos(x) x = cos(y) −1 to +1 0 ≤ y ≤ π
arctangent y = arctan(x) x = tan(y) all −π/2 < y < π/2
arccotangent y = arccot(x) x = cot(y) all 0 < y < π
arcsecant y = arcsec(x) x = sec(y) −∞ to −1 or 1 to ∞ 0 ≤ y < π/2 or π/2 < y ≤ π
arccosecant y = arccsc(x) x = csc(y) −∞ to −1 or 1 to ∞ −π/2 ≤ y < 0 or 0 < y ≤ π/2
TRIGONOMETRIC RATIOS FOR ACUTE ANGLES
We use a right angled triangle to consider the Trig. Ratio and we remember that
the Ratio of Corresponding Sides in Similar Triangles remains constant. Given a
triangle ABC we denote the lengths of the sides to be a,b and c.
There are 6 Ratios and are defined as follows: 3 MAJOR and 3 MINOR
self-test 12: triangle b = x, c = 2*x, a = ?
sin B = ?, cos A = ?
In each right-angled triangle ABC, with A as right angle, we have
sin(B) = b/a
cos(B) = c/a
cos(C) = b/a
sin(C) = c/a
tan(B) = b/c
tan(C) = c/b
Right angle triangle Pythagoras:
c2 = a2 + b2
General triangle Pythagoras:
‘Law of cosines’
c2 = a2 + b2 - 2 a b cos (C)
Example, triangle with all angles 60 degree, a = 1, c = ?
c2 = 12 + 12 – 2 * 1 *1 * cos(60) = 1 + 1 – 2 * 0.5 = 1
Rules for trigonometric functions:
Identities:
Sum and difference identities
self-test 13: use triangle from self
test 12 and confirm formula for sin(A+B)
Double-angle identities
Half-angle identities
Self-test 14: test cosC formula for triangle
from self test 12
Law of tangents
http://press.princeton.edu/books/maor/trig delights , Eli Maor